A335631 a(n) is the number of decompositions of F(n,1) into disjoint unions of F(j,k) and G(q,r) where F(j,k) is the set of numbers { i*k, 0 <= i <= j } and where G(q,r) is the set of numbers { (2*p-1)*r, 1 <= p <= q }.

1, 3, 6, 11, 17, 32, 47, 66, 105, 162, 198, 376, 451, 634, 1131, 1405, 1576, 2487, 2761, 4216, 7499, 9207, 9931, 19328, 25621, 30701, 48795, 67006, 70841, 122397, 128833, 143082, 270155, 303385, 493906, 761627, 789651, 877486, 1715085, 2595269, 2679136, 4665660

Offset

1,2

Links

Table of n, a(n) for n=1..42.

M. Znojil, Non-Hermitian N-state degeneracies: unitary realizations via antisymmetric anharmonicities, arXiv:2010.15014 [quant-ph], 2020.

M. Znojil, Quantum phase transitions mediated by clustered non-Hermitian degeneracies, arXiv:2102.12272 [quant-ph], 2021.

M. Znojil, Quantum phase transitions mediated by clustered non-Hermitian degeneracies, Physical Review E 103 (2021), 032120.

Miloslav Znojil, Bose-Einstein condensation processes with nontrivial geometric multiplicites realized via PT-symmetric and exactly solvable linear-Bose-Hubbard building blocks, arXiv:2108.07110 [quant-ph], 2021.

Example

F(n,1) are the sets {0,1}, {0,1,2}, {0,1,2,3}, {0,1,2,3,4}, ...
F(n,2) are the sets {0,2}, {0,2,4}, {0,2,4,6}, {0,2,4,6,8}, ...
F(n,3) are the sets {0,3}, {0,3,6}, {0,3,6,9}, {0,3,6,9,12}, ...
etc., and
G(n,1) are the sets {1}, {1,3}, {1,3,5}, {1,3,5,7}, ...
G(n,2) are the sets {2}, {2,6}, {2,6,10}, {2,6,10,14}, ...
G(n,3) are the sets {3}, {3,9}, {3,9,15}, {3,9,15,21}, ...
etc.
a(2) = 3 because there are three decompositions of F(2,1) = {0,1,2}: the trivial F(2,1), F(1,2) + G(1,1) = {0,2} + {1}, and F(1,1) + G(1,2) = {0,1} + {2}.
The a(3) = 6 decompositions of {0,1,2,3} are:
  {{0,1,2,3}},
  {{0,1,2}, {3}},
  {{0,1}, {2}, {3}},
  {{0,2}, {1,3}},
  {{0,2}, {1}, {3}},
  {{0,3}, {1}, {2}}.

Prog

(PARI)
tiles(S, t, w)={((i, b)->w(b) + sum(i=1, i, if(!bitnegimply(S[i], b), self()(i-1, b-S[i]))))(#S, t)}
Q(j, k, t)={sum(i=0, j-1, 1<<(t+i*k))}
S(n)={concat(concat(vector(n\2+1, k, vector(n\k-1, j, Q(j+2, k, 0)))), concat(vector(n\3+1, k, vector((n\k-1)\2, j, Q(j+1, 2*k, k)))))}
a(n)={tiles(S(n), Q(n+1, 1, 0), b->if(bittest(b, 0), hammingweight(b)-1, 1))} \\ Andrew Howroyd, Oct 29 2020

Crossrefs

Cf. A336739 (odd-number-sets decomposition).

Sequence in context: A109471 A279032 A124454 * A013932 A302550 A310117

Adjacent sequences: A335628 A335629 A335630 * A335632 A335633 A335634

Keyword

nonn

Author

Miloslav Znojil, Oct 03 2020

Extensions

a(10)-a(42) from Andrew Howroyd, Oct 29 2020

Status

approved